Descartes' Rule of Signs Calculator

About this calculator

Descartes' Rule of Signs counts the sign changes between consecutive nonzero coefficients of a polynomial, ordered from highest to lowest degree, to bound the number of positive real roots — the actual count is that number of sign changes, or fewer by an even number. Substituting x with −x and repeating the count gives the equivalent bound for negative real roots, all without needing to actually solve the polynomial.

This calculator applies the rule to your entered coefficients, a technique taught in algebra and precalculus courses as a quick way to narrow down how many real roots to expect before applying numerical methods or the rational root theorem. It's especially useful for sanity-checking a graphing calculator's output or for problems where finding exact roots is hard but knowing how many to look for is enough.